Tracer Diffusion in Low Dimensional Lattice Gases: Probing the Einstein Relation

1. Tracer Diffusion in Low Dimensional Lattice Gases: Probing the Einstein Relation Gleb OSHANIN Physique Théorique de la Matière Condensée, Paris 6/ CNRS May 11, 2007

2. Tracer Diffusion in a 1D Lattice Gas (1D exclusion process) p q Tracer particle (TP), position X(t) p and q are the transition probabilities Particles of the lattice gas, present at mean density n Mean-square displacement of the TP in the symmetric case p=q (Harris, 1965): i.e. diffusivity : We have shown (Burlatsky, Oshanin and Moreau, 1992) that in the totally asymmetric case, when p = 0 and q = 1: Hence, mobility follows: A question (due to J.L.Lebowitz):

3. 1.Directed walk in a one-dimensional lattice gas, with S Burlatsky, A Mogutov and M Moreau, Phys. Lett. A 166, 230 (1992) 2. Motion of a driven tracer particle in a one-dimensional lattice gas, with S F Burlatsky, M Moreau and W P Reinhardt, Phys. Rev. E 54, 3165 (1996) 3. Dynamics of a driven probe molecule in a liquid monolayer, with J De Coninck and M Moreau, Europhys. Lett. 38, 527 (1997) 4. Biased diffusion in a one-dimensional adsorbed monolayer, with O Bénichou, A M Cazabat, A Lemarchand and M Moreau, J. Stat. Phys. 97, 351 (1999) 5. Directed random walk in adsorbed monolayer, with O Bénichou, A M Cazabat and M Moreau, Physica A 272, 56 (1999) 6. Stokes formula and density perturbances for driven tracer diffusion in an adsorbed monolayer, with O Bénichou, A M Cazabat, J De Coninck and M Moreau, Phys. Rev. Lett. 84, 511 (2000) 7. Generalized model for dynamic percolation, with O Bénichou, J Klafter and M Moreau, Phys. Rev. E 62, 3327 (2000) 8. Phase boundary dynamics in a one-dimensional non-equilibrium lattice gas, with J De Coninck, M Moreau and S F Burlatsky, in: Instabilities and Non-Equilibrium Structures VII, ed. E Tirapegui, (Kluwer Academic Pub., Dordrecht) 9.Force-velocity relation and density profiles for biased diffusion in adsorbed monolayers, with O Bénichou, A M Cazabat, J De Coninck and M Moreau, Phys. Rev. B 63, 235413 (2001) 10. Intrinsic friction of adsorbed monolayers, with O Benichou, A M Cazabat, J De Coninck and M Moreau, J. Phys. C 13, 4835 (2001) 11.Ultra-slow vacancy-mediated tracer diffusion in two-dimensions: The Einstein relation verified, with O Bénichou, Phys. Rev. E 66, 031101 (2002) 12. The atomic slide puzzle: self-diffusion of an impure atom, with O Bénichou, Phys. Rev. E 64, R020103 (2001) 13. Biased tracer diffusion in hard-core lattice gases. Some notes on the validity of the Einstein relation. To appear in Instabilities and Non-Equilibrium structures. Kluwer

4. Why are we interested in tracer diffusion? 1. probe frictional force exerted on tracer particle by the medium – access to intrinsic frictional properties/viscosity 2. emphasize cooperative behavior/backflow effects 3. verify the Einstein relation.

5. Summary 1. Notations and analytical approach (for d dimensional case) 2. Biased tracer diffusion in 1D systems. Einstein relation. Effects of non-equilib-rium density (shock propagation) and reservoir (vapor phase). 3. Biased tracer diffusion in 2D lattice gas. Einstein relation. Density profiles. 4. Single-vacancy mediated tracer diffusion. Einstein relation.

6. General Model VAPOR PHASE E Adsorption site SOLID Adsorbed hard-core particles may desorb back to the vapor phase or move randomly on a d- dimensional lattice: Langmuir + diffusion The tracer particle is subject to external field E, which favors its jumps in a preferential direction . Applications: intrinsic friction of adsorbed monolayers; viscosity of stagnant layers - Force/Velocity relation ? - Frictional force exerted on the TP ? - Density profiles as seen from the TP ?

7. VAPOR t*/f t*/g E t Adsorption site t* SOLID s Model Parameters Stationary density in absence of the tracer: = basis vector of a hypercubic lattice, E is parallel to e1 = probability that theTP jumps in the direction Limiting situations - Absence of bias : E = 0 - Absence of particles exchanges with the reservoir (vapor phase) . Conserved particle number : f 0, g 0, f/(f+g) fixed

8. Evolution Equations Given particles configuration = (X,{h (r)}) X = position of the tracer particle h (r)=occupation number of siter =0 ou 1 Evolution of P(X,{h(r)}) = Master Equation Dynamics of the lattice gas particles Dynamics of the tracer particle Desorption Adsorption = diffusion process Kawasaki = adsorption/desorption processes Glauber

9. Density profiles = pair correlation function Velocity of the tracer particle Coupled problem Evolution equations for the density profiles - deduced from the Master Equation - are coupled to third order correlations - depend explicitly on the tracer particle velocity Equations are not closed Coupled problem

10. Approach A priori Resolution of an infinite hierarchy of coupled differential equations (in finite differences) Approximation Decoupling Stationary regime Solution at long times: stationary profiles and constant velocity Problem Density profiles = solution of non-linear differential equations + boundaryconditions (sites adjacent to tracer) supposing that its velocity is known. Then, we obtain closed-form equation for the tracer velocity. Very similar problem = Stefan problem or directional solidification.

11. Our results can be checked against: In absence of external bias and vapor phase Symmetric tracer diffusion in a lattice gas (density n ) executes a correleted random walk • - limitn 0 (isolated tracer) : simple random walk • - limite n 1 (density of lacunes  0) • Bardeen et Herring, Brummelhuis et Hilhorst … • n arbitrary: a genuine N body problem • Nakazato et Kitahara: approximate • calculation of DNK • Tahir-Kheli etElliot : • DNK exact when n  1 • n  0 • Kehr et Binder : numerical simulations • DNK =very good approximation n

12. One dimensional systems In absence of the vapor phase (reservoir) E Burlatsky, Oshanin, Moreau, Reinhard (1996), Landim and Olla (1998) Anomalous behavior : For bE << 1 Einstein relation holds exactly!!! Density profile – step-function

13. One dimensional systems In absence of the vapor phase Initial nonhomogeneous (S-shape) density profile E High density phase Low density phase E - is an effective tension (within the SOS model approximation) of the interface between the high- and low-density phases

14. Three different regimes: I. The high-density phase expands when when II. The high-density phase retracts (dewets) when

15. III. The low- and high-density phases are in equilibrium with each other when Here, Mobility (not zero!) Assuming that the Einstein Relation holds

16. One dimensional systems In presence of the vapor phase E Existence of an additional dimension   - i.e. possibility of particles exchanges with the reservoir (no strict conservation of particles number)  - does it reduce the effect of one-dimensional confinement? ?

17. Limit of small E : friction coefficient z Defined by a Stokes-type formula : where : - mean-field-type contribution: (Langmuir) Effective frequency of jumps = bare frequency of jumps  fraction of successful jumps - cooperative effects (formation of stationary density profiles around stationary moving TP)contribution: where

18. Stationary velocity General case Stationary TP Velocity as a function of f for g=0.3 ; 0.5 ; 0.8 ; p=0.6 - MC simulations • Analytical solution • (discrete chain) Very good agreement

19. E=0. Diffusion Coefficient If the Einstein Relation holds, we should obtain where

20. Stationary density profiles around the TP for f=0.1 ; g=0.3 ; p1=0.98 : analytic solution based on the decoupling : results of numeric simulations Stationary Density Profiles as seen from the TP traffic jam in front of + depletion past theTP kl l Very good agreement

21. Two-Dimensional Systems Situation bidimensionnelle Discrete-space decoupled evolution equations are solved by resorting to the generating function technique: where = deviation from the unperturbed value = and (l1,l2) = site of a square lattice Allows for straightforward calculation of all pertinent physical properties Calculation of H Integral characteristics of density profiles: Global compensation of inhomogeneities of the density profiles

22. Inversion of H with respect to w2 Analysis of the analytical behaviorof asymptotical behavior of the density profiles at a large distance from the TP - In front of the tracer particle (exponential relaxation to the unperturbed value) - Past the tracer particle 1)Non conserved particle number (vapor phase) with 2)Conserved particle number (no vapor phase) : algebraic relaxation! Strong memory effects past (in the wake of) the tracer

23. Large E limit The tracer particle executes a totally directed random walk Explicit solution Density profiles at a finite distance from the TP, In the limit of high diffusion coefficient of the monolayer particles E

24. Small E limit Friction coefficient z Mean-field with where P(r;x) : Green’s function of standard random walk on a two-dimensional square lattice Particular case : conserved particle number and E = 0 Einstein relation yields: = result of Nakazato and Kitahara = exactin the vicinity of n=0 andn=1and very good approximation for intermediate densities

25. Haute densité High Density Limit: The Atomic Slide Puzzle Motivations -  Natural  limit of the previous model - validity of the Einstein relation ? Model with a single vacancy, discrete time t : tracer particle : vacancy • Combines two interesting effects : • - correlation between successive jumps of the tracer particle • - broad distribution of waiting times : very slow dynamics

26. Experimental relevance of the single-vacancy model At room temperature one atom out of 6 x 109 is missing = one vacancy per terrace of width 2 x 105 A. From Raoul van Gastel Ph.D. Thesis, Cammerlingh Onnes Lab, Leiden University, 2001

27. Red – Cu atoms, Yellow – embedded Indium atoms STM Images, courtesy of Raoul van Gastel

28. How the motion proceeds Continuous reshuffling of the surface = Atomic Slide Puzzle

29. Validity of the Einstein Relation A reminder : in absence of external bias Brummelhuis and Hilhorst - Anomalous law : - Diffusion coefficient at long times : Here : in presence of external bias acting on the tracer Exact result for the tracer mobility : Do we have ? • Answer is non-trivial a priori since : • - Generalized Einstein Realtion • - Strong effects of temporal trapping

35. Results in presence of external bias Asymptotically exact results General Force/Velocity Relation : where Limit of vanishingly small bias (E 0) : Einstein Relation holds! Generalization for finite vacancy concentration. Limit density  1 : = result of the previous model of adsorbed monolayer

41. Conclusions • - For several models we have evaluated general • force/velocity relations • - Have demonstrated the validity of the generalized Einstein Relation • - Explicit expressions for frictioncoefficient : - Determined the density profiles: Effects of traffic jams and depletion in the wake - Particular case of conserved particle number : strong memory effects past the tracer particle – algebraic tails - High Density Limit : exact results Perspectives - realistic interactions between the particles - many tracer effects, overlap of the perturbed regions